Model Checking Spatial Logics for Closure Spaces

Spatial aspects of computation are becoming increasingly relevant in Computer Science, especially in the field of collective adaptive systems and when dealing with systems distributed in physical space. Traditional formal verification techniques are well suited to analyse the temporal evolution of programs; however, properties of space are typically not taken into account explicitly. We present a topology-based approach to formal verification of spatial properties depending upon physical space. We define an appropriate logic, stemming from the tradition of topological interpretations of modal logics, dating back to earlier logicians such as Tarski, where modalities describe neighbourhood. We lift the topological definitions to the more general setting of closure spaces, also encompassing discrete, graph-based structures. We extend the framework with a spatial surrounded operator, a propagation operator and with some collective operators. The latter are interpreted over arbitrary sets of points instead of individual points in space. We define efficient model checking procedures, both for the individual and the collective spatial fragments of the logic and provide a proof-of-concept tool

Distance Closures on Complex Networks

To expand the toolbox available to network science, we study the isomorphism between distance and Fuzzy (proximity or strength) graphs. Distinct transitive closures in Fuzzy graphs lead to closures of their isomorphic distance graphs with widely different structural properties. For instance, the All Pairs Shortest Paths (APSP) problem, based on the Dijkstra algorithm, is equivalent to a metric closure, which is only one of the possible ways to calculate shortest paths. Understanding and mapping this isomorphism is necessary to analyse models of complex networks based on weighted graphs. Any conclusions derived from such models should take into account the distortions imposed on graph topology when converting proximity/strength into distance graphs, to subsequently compute path length and shortest path measures. We characterise the isomorphism using the max-min and Dombi disjunction/conjunction pairs. This allows us to: (1) study alternative distance closures, such as those based on diffusion, metric, and ultra-metric distances; (2) identify the operators closest to the metric closure of distance graphs (the APSP), but which are logically consistent; and (3) propose a simple method to compute alternative distance closures using existing algorithms for the APSP. In particular, we show that a specific diffusion distance is promising for community detection in complex networks, and is based on desirable axioms for logical inference or approximate reasoning on networks; it also provides a simple algebraic means to compute diffusion processes on networks. Based on these results, we argue that choosing different distance closures can lead to different conclusions about indirect associations on network data, as well as the structure of complex networks, and are thus important to consider

Quotient on some Generalizations of topological group

في هذا البحث ، تم تعريف بعض التعميمات للمجموعة التبولوجية وهي المجموعة التبولوجية - α ، والمجموعة التبولوجية - ب ، والمجموعة التبولوجية - β  مع أمثلة توضيحية. بالإضافة إلى ذلك ، تم تعريف المجموعة التبولوجية للشواء فيما يتعلق بالشواية. فيما بعد ، تم تداول حاصل قسمة تعميمات المجموعة التبولوجية في مجموعة تبولوجية - p معينة. علاوة على ذلك ، تمت مناقشة نموذج النظام الروبوتي الذي يعتمد على حاصل المجموعة التبولوجية – p.In this paper, we define some generalizations of topological group namely -topological group, -topological group and -topological group with illustrative examples. Also, we define grill topological group with respect to a grill. Later, we deliberate the quotient on generalizations of topological group in particular -topological group. Moreover, we model a robotic system which relays on the quotient of -topological group

06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality

From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

On Quasi-Isometry of Threshold-Based Sampling

The problem of isometry for threshold-based sampling such as integrate-and-fire (IF) or send-on-delta (SOD) is addressed. While for uniform sampling the Parseval theorem provides isometry and makes the Euclidean metric canonical, there is no analogy for threshold-based sampling. The relaxation of the isometric postulate to quasi-isometry, however, allows the discovery of the underlying metric structure of threshold-based sampling. This paper characterizes this metric structure making Hermann Weyl's discrepancy measure canonical for threshold-based sampling.Comment: submitted to IEEE Transactions on Signal Processin

Annotation Scaffolds for Object Modeling and Manipulation

We present and evaluate an approach for human-in-the-loop specification of shape reconstruction with annotations for basic robot-object interactions. Our method is based on the idea of model annotation: the addition of simple cues to an underlying object model to specify shape and delineate a simple task. The goal is to explore reducing the complexity of CAD-like interfaces so that novice users can quickly recover an object's shape and describe a manipulation task that is then carried out by a robot. The object modeling and interaction annotation capabilities are tested with a user study and compared against results obtained using existing approaches. The approach has been analyzed using a variety of shape comparison, grasping, and manipulation metrics, and tested with the PR2 robot platform, where it was shown to be successful.Comment: 31 pages, 46 Figure

Spatial Logics and Model Checking for Medical Imaging (Extended Version)

Recent research on spatial and spatio-temporal model checking provides novel image analysis methodologies, rooted in logical methods for topological spaces. Medical Imaging (MI) is a field where such methods show potential for ground-breaking innovation. Our starting point is SLCS, the Spatial Logic for Closure Spaces -- Closure Spaces being a generalisation of topological spaces, covering also discrete space structures -- and topochecker, a model-checker for SLCS (and extensions thereof). We introduce the logical language ImgQL ("Image Query Language"). ImgQL extends SLCS with logical operators describing distance and region similarity. The spatio-temporal model checker topochecker is correspondingly enhanced with state-of-the-art algorithms, borrowed from computational image processing, for efficient implementation of distancebased operators, namely distance transforms. Similarity between regions is defined by means of a statistical similarity operator, based on notions from statistical texture analysis. We illustrate our approach by means of two examples of analysis of Magnetic Resonance images: segmentation of glioblastoma and its oedema, and segmentation of rectal carcinoma

Using GPS technology to quantify human mobility, dynamic contacts and infectious disease dynamics in a resource-poor urban environment.

Empiric quantification of human mobility patterns is paramount for better urban planning, understanding social network structure and responding to infectious disease threats, especially in light of rapid growth in urbanization and globalization. This need is of particular relevance for developing countries, since they host the majority of the global urban population and are disproportionally affected by the burden of disease. We used Global Positioning System (GPS) data-loggers to track the fine-scale (within city) mobility patterns of 582 residents from two neighborhoods from the city of Iquitos, Peru. We used ∼2.3 million GPS data-points to quantify age-specific mobility parameters and dynamic co-location networks among all tracked individuals. Geographic space significantly affected human mobility, giving rise to highly local mobility kernels. Most (∼80%) movements occurred within 1 km of an individual's home. Potential hourly contacts among individuals were highly irregular and temporally unstructured. Only up to 38% of the tracked participants showed a regular and predictable mobility routine, a sharp contrast to the situation in the developed world. As a case study, we quantified the impact of spatially and temporally unstructured routines on the dynamics of transmission of an influenza-like pathogen within an Iquitos neighborhood. Temporally unstructured daily routines (e.g., not dominated by a single location, such as a workplace, where an individual repeatedly spent significant amount of time) increased an epidemic's final size and effective reproduction number by 20% in comparison to scenarios modeling temporally structured contacts. Our findings provide a mechanistic description of the basic rules that shape human mobility within a resource-poor urban center, and contribute to the understanding of the role of fine-scale patterns of individual movement and co-location in infectious disease dynamics. More generally, this study emphasizes the need for careful consideration of human social interactions when designing infectious disease mitigation strategies, particularly within resource-poor urban environments

Dirac Triples for Unital AF Algebras

For a unital AF algebra A, we construct a family of triples (A, H, D) where A is represented faithfully on the Hilbert space H and D is an unbounded self-adjoint operator on H. These triples have the same properties as spectral triples except for the compact resolvent condition, so we call them Dirac triples. They serve as a generalization of Pearson-Bellissard spectral triples for an ultrametric Cantor set corresponding to choice functions. Pearson and Bellissard showed that the underlying ultrametric can be recovered by considering spectral triples associated to all choice functions. We obtain an analogue for unital AF algebras: the supremum of the Connes spectral distances induced by a large family of Dirac triples from our construction coincides with a generalized version of the Aguilar seminorm, which is a Leibniz Lip-norm for a unital AF algebra. Moreover, the convergence result of Aguilar is retained: equipped with the generalized Aguilar seminorm, a unital AF algebra is the limit of its defining finite-dimensional subalgebras for the quantum Gromov-Hausdorff propinquity

A digital analogue of the Jordan curve theorem

AbstractWe study certain closure operations on Z2, with the aim of showing that they can provide a suitable framework for solving problems of digital topology. The Khalimsky topology on Z2, which is commonly used as a basic structure in digital topology nowadays, can be obtained as a special case of the closure operations studied. By proving an analogy of the Jordan curve theorem for these closure operations, we show that they provide a convenient model of the real plane and can therefore be used for studying topological and geometric properties of digital images. We also discuss some advantages of the closure operations investigated over the Khalimsky topology